First of all there was no history in the Tarim in 2000 BC, just prehistory, so we must rely in archaeology, so then you might deliver here which archaeological proofs are linking the Altai with the Tarim desert (weapons? pots?)
Well, that's a opinion of some Russian/Soviet archeologists as Danilenko, Safronov etc.
Along with this, the Proto-Tocharian possible to associate with the Karasuk culture.
https://en.wikipedia.org/wiki/Karasuk_culture
On the whole, only two culture can be Tocharians. Afanasevo or Karasuk.
For example Russian archaeologist Klein believes that Afanasievo and Karasuk culture (about Karasuk confirmed by genetics) does not come from Yamnaya culture. He also writes that Karasuk culture more suited to Tocharian migrations. Karasuk type items were found in Xinjiang, where lives Tochars. Tocharians have Finno-Ugric substratum in their language. And Finno-Ugric have substratum from Tocharian ( honey 'mete' (fu) from 'met' (tochar) etc). Hence the conclusion that they were walking along the forest belt, far from steppe.